how do you find reference angles in the four quadrants

To find reference angles in the four quadrants, follow these steps:

1. Determine the given angle's quadrant: Each quadrant has a specific range of angles. The first quadrant is from 0 to 90 degrees (0°-90°), the second quadrant is from 90 to 180 degrees (90°-180°), the third quadrant is from 180 to 270 degrees (180°-270°), and the fourth quadrant is from 270 to 360 degrees (270°-360°).

2. Determine the reference angle for angles in the first quadrant: Since the reference angle is the acute angle formed between the terminal side of the given angle and the x-axis, the reference angle in the first quadrant is simply the given angle itself.

3. Determine the reference angle for angles in the other three quadrants: Subtract the given angle from the closest multiple of 180 degrees.

- In the second quadrant, subtract the given angle from 180 degrees.
- In the third quadrant, subtract the given angle from 180 degrees.
- In the fourth quadrant, subtract the given angle from 360 degrees.

The resulting value will be the reference angle for angles in the second, third, and fourth quadrants.

Note: If the given angle is already a multiple of 90 degrees, its reference angle is 0 degrees.

For example:
- Given angle: 150 degrees in the second quadrant
Reference angle = 180 degrees - 150 degrees = 30 degrees

- Given angle: -210 degrees in the third quadrant
Reference angle = 180 degrees - (-210 degrees) = 390 degrees (Add 360 degrees to make it positive) = 30 degrees

By following these steps, you can find the reference angles for angles in the four quadrants.